 Solution

For $0<r<1$, start at any $x\in \{0,\dots ,n\}$. Propose $Y$ uniformly from this same finite set and accept it with probabilityOtherwise retain $x$. The uniform proposal is symmetric, so this is the Metropolis–Hastings algorithm with target $\pi (\cdot \mid r)$. Detailed balance follows because accepted probability flow is proportional to $\min \{\pi (x),\pi (Y)\}$. The chain is irreducible and has positive self-transition probability, hence converges to that binomial equilibrium law. At the degenerate parameters zero or one, the target is simply a point mass and can be simulated as a constant chain.